arXiv subjects

Shikher Sharma

Publications and source records attributed to Shikher Sharma.

3 recordsLinked to original sources

arXiv2026

A Bregman Regularized Proximal Point Method for Solving Equilibrium Problems on Hadamard Manifolds

In this paper we develop a Bregman regularized proximal point algorithm for solving monotone equilibrium problems on Hadamard manifolds. It has been shown that the regularization term induced by a Bregman function is, in general, nonconvex on Hadamard manifolds unless the curvature is zero. Nevertheless, we prove that the proposed Bregman regularization scheme does converge to a solution of the equilibrium problem on Hadamard manifolds in the presence of a strong assumption on the convexity of the set formed by the regularization term. Moreover, we employ a coercivity condition on the Bregman function which is weaker than those typically assumed in the existing literature on Bregman regularization. Numerical experiments on illustrative examples demonstrate the practical effectiveness of our proposed method.

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arXiv2025

Efficient Douglas-Rachford Methods on Hadamard Manifolds with Applications to the Heron Problems

Our interest lies in developing some efficient methods for minimizing the sum of two geodesically convex functions on Hadamard manifolds, with the aim to enhance the convergence of the Douglas-Rachford algorithm in Hadamard manifolds. Specifically, we propose two types of algorithms: inertial and non-inertial algorithms. The convergence analysis of both algorithms is provided under suitable assumptions on algorithmic parameters and the geodesic convexity of the objective functions. This convergence analysis is based on fixed-point theory for nonexpansive operators. We also study the convergence rates of these two methods. Additionally, we introduce parallel Douglas-Rachford type algorithms for minimizing functionals containing multiple summands with applications to the generalized Heron problem on Hadamard manifolds. To demonstrate the effectiveness of the proposed algorithms, we present some numerical experiments for the generalized Heron problems.

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arXiv2025

A Modified Dai-Liao Spectral Conjugate Gradient Method with an Application to Signal Processing

We propose and study a variant of the Dai-Liao spectral conjugate gradient method, developed through an analysis of eigenvalues and inspired by a modified secant condition. We show that our proposed method is globally convergent for general nonlinear functions under standard assumptions. By incorporating the new secant condition and a quasi-Newton direction, we introduce updated spectral parameters. These changes ensure that the resulting search direction satisfies the sufficient descent property without relying on any line search. Numerical experiments show that the proposed algorithm performs better than several existing methods in terms of convergence speed and computational efficiency. Its effectiveness is further demonstrated through an application to signal processing.

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